Next Article in Journal
Special Numbers and Polynomials Including Their Generating Functions in Umbral Analysis Methods
Next Article in Special Issue
Interval Neutrosophic Sets with Applications in BCK/BCI-Algebra
Previous Article in Journal
Graphs in an Intuitionistic Fuzzy Soft Environment
Previous Article in Special Issue
Multi-Attribute Decision-Making Method Based on Neutrosophic Soft Rough Information
Article Menu

Export Article

Axioms 2018, 7(2), 21; doi:10.3390/axioms7020021

Article
Cross Entropy Measures of Bipolar and Interval Bipolar Neutrosophic Sets and Their Application for Multi-Attribute Decision-Making
1
Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, P.O.-Narayanpur, District–North 24 Parganas, West Bengal 743126, India
2
Department of Mathematics, Patipukur Pallisree Vidyapith, Patipukur, Kolkata 700048, India
3
Mathematics & Science Department, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA
4
Department of Electrical and Information Engineering, Shaoxing University, 508 Huancheng West Road, Shaoxing 312000, China
*
Author to whom correspondence should be addressed.
Received: 7 January 2018 / Accepted: 22 March 2018 / Published: 24 March 2018

Abstract

:
The bipolar neutrosophic set is an important extension of the bipolar fuzzy set. The bipolar neutrosophic set is a hybridization of the bipolar fuzzy set and neutrosophic set. Every element of a bipolar neutrosophic set consists of three independent positive membership functions and three independent negative membership functions. In this paper, we develop cross entropy measures of bipolar neutrosophic sets and prove their basic properties. We also define cross entropy measures of interval bipolar neutrosophic sets and prove their basic properties. Thereafter, we develop two novel multi-attribute decision-making strategies based on the proposed cross entropy measures. In the decision-making framework, we calculate the weighted cross entropy measures between each alternative and the ideal alternative to rank the alternatives and choose the best one. We solve two illustrative examples of multi-attribute decision-making problems and compare the obtained result with the results of other existing strategies to show the applicability and effectiveness of the developed strategies. At the end, the main conclusion and future scope of research are summarized.
Keywords:
neutrosophic set; bipolar neutrosophic set; interval bipolar neutrosophic set; multi-attribute decision-making; cross entropy measure

1. Introduction

Shannon and Weaver [1] and Shannon [2] proposed the entropy measure which dealt formally with communication systems at its inception. According to Shannon and Weaver [1] and Shannon [2], the entropy measure is an important decision-making apparatus for computing uncertain information. Shannon [2] introduced the concept of the cross entropy strategy in information theory.
The measure of a quantity of fuzzy information obtained from a fuzzy set or fuzzy system is termed fuzzy entropy. However, the meaning of fuzzy entropy is quite different from the classical Shannon entropy because it is defined based on a nonprobabilistic concept [3,4,5], while Shannon entropy is defined based on a randomness (probabilistic) concept. In 1968, Zadeh [6] extended the Shannon entropy to fuzzy entropy on a fuzzy subset with respect to the concerned probability distribution. In 1972, De Luca and Termini [7] proposed fuzzy entropy based on Shannon’s function and introduced the axioms with which the fuzzy entropy should comply. Sander [8] presented Shannon fuzzy entropy and proved that the properties sharpness, valuation, and general additivity have to be imposed on fuzzy entropy. Xie and Bedrosian [9] proposed another form of total fuzzy entropy. To overcome the drawbacks of total entropy [8,9], Pal and Pal [10] introduced hybrid entropy that can be used as an objective measure for a proper defuzzification of a certain fuzzy set. Hybrid entropy [10] considers both probabilistic entropies in the absence of fuzziness. In the same study, Pal and Pal [10] defined higher-order entropy. Kaufmann and Gupta [11] studied the degree of fuzziness of a fuzzy set by a metric distance between its membership function and the membership function (characteristic function) of its nearest crisp set. Yager [12,13] introduced a fuzzy entropy card as a fuzziness measure by observing that the intersection of a fuzzy set and its complement is not the void set. Kosko [14,15] studied the fuzzy entropy of a fuzzy set based on the fuzzy set geometry and distances between them. Parkash et al. [16] proposed two new measures of weighted fuzzy entropy.
Burillo and Bustince [17] presented an axiomatic definition of an intuitionistic fuzzy entropy measure. Szmidt and Kacprzyk [18] developed a new entropy measure based on a geometric interpretation of the intuitionistic fuzzy set (IFS). Wei et al. [19] proposed an entropy measure for interval-valued intuitionistic fuzzy sets (IVIFSs) and employed it in pattern recognition and multi criteria decision-making (MCDM). Li [20] presented a new multi-attribute decision-making (MADM) strategy combining entropy and technique for order of preference by similarity to ideal solution (TOPSIS) in the IVIFS environment.
Shang and Jiang [21] developed cross entropy in the fuzzy environment. Vlachos and Sergiadis [22] presented intuitionistic fuzzy cross entropy by extending fuzzy cross entropy [21]. Ye [23] proposed a new cross entropy in the IVIFS environment and developed an optimal decision-making strategy. Xia and Xu [24] defined a new entropy and a cross entropy and presented multi-attribute group decision-making (MAGDM) strategy in the IFS environment. Tong and Yu [25] defined cross entropy in the IVIFS environment and employed it to solve MADM problems.
Smarandache [26] introduced the neutrosophic set, which is a generalization of the fuzzy set [27] and intuitionistic fuzzy set [28]. The single-valued neutrosophic set (SVNS) [29], an instance of the neutrosophic set, has caught the attention of researchers due to its applicability in decision-making [30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61], conflict resolution [62], educational problems [63,64], image processing [65,66,67], cluster analysis [68,69], social problems [70,71], etc.
Majumdar and Samanta [72] proposed an entropy measure and presented an MCDM strategy in the SVNS environment. Ye [73] defined cross entropy for SVNS and proposed an MCDM strategy which bears undefined phenomena. To overcome the undefined phenomena, Ye [74] defined improved cross entropy measures for SVNSs and interval neutrosophic sets (INSs) [75], which are straightforward symmetric, and employed them to solve MADM problems. Since MADM strategies [73,74] are suitable for single-decision-maker-oriented problems, Pramanik et al. [76] defined NS-cross entropy and developed an MAGDM strategy which is straightforward symmetric and free from undefined phenomena and suitable for group decision making problem. Şahin [77] proposed two techniques to convert the interval neutrosophic information to single-valued neutrosophic information and fuzzy information. In the same study, Şahin [77] defined an interval neutrosophic cross entropy measure by utilizing two reduction methods and an MCDM strategy. Tian et al. [78] developed a transformation operator to convert interval neutrosophic numbers to single-valued neutrosophic numbers and defined cross entropy measures for two SVNSs. In the same study, Tian et al. [78] developed an MCDM strategy based on cross entropy and TOPSIS [79] where the weight of the criterion is incomplete. Tian et al. [78] defined a cross entropy for INSs and developed an MCDM strategy based on the cross entropy and TOPSIS. The MCDM strategies proposed by Sahin [77] and Tian et al. [78] are applicable for a single decision maker only. Therefore, multiple decision-makers cannot participate in the strategies in [77,78]. To tackle the problem, Dalapati et al. [80] proposed IN-cross entropy and weighted IN-cross entropy and developed an MAGDM strategy.
Deli et al. [81] proposed bipolar neutrosophic set (BNS) by hybridizing the concept of bipolar fuzzy sets [82,83] and neutrosophic sets [26]. A BNS has two fully independent parts, which are positive membership degree T+ [0, 1], I+ [0, 1], F+ [0, 1], and negative membership degree T [−1, 0], I [−1, 0], F [−1, 0], where the positive membership degrees T+, I+, F+ represent truth membership degree, indeterminacy membership degree, and false membership degree, respectively, of an element and the negative membership degrees T, I, F represent truth membership degree, indeterminacy membership degree, and false membership degree, respectively, of an element to some implicit counter property corresponding to a BNS. Deli et al. [81] defined some operations, namely, score, accuracy, and certainty functions, to compare BNSs and provided some operators in order to aggregate BNSs. Deli and Subas [84] defined a correlation coefficient similarity measure for dealing with MCDM problems in a single-valued bipolar neutrosophic setting. Şahin et al. [85] proposed a Jaccard vector similarity measure for MCDM problems with single-valued neutrosophic information. Uluçay et al. [86] introduced a Dice similarity measure, weighted Dice similarity measure, hybrid vector similarity measure, and weighted hybrid vector similarity measure for BNSs and established an MCDM strategy. Dey et al. [87] investigated a TOPSIS strategy for solving multi-attribute decision-making (MADM) problems with bipolar neutrosophic information where the weights of the attributes are completely unknown to the decision-maker. Pramanik et al. [88] defined projection, bidirectional projection, and hybrid projection measures for BNSs and proved their basic properties. In the same study, Pramanik et al. [88] developed three new MADM strategies based on the proposed projection, bidirectional projection, and hybrid projection measures with bipolar neutrosophic information. Wang et al. [89] defined Frank operations of bipolar neutrosophic numbers (BNNs) and proposed Frank bipolar neutrosophic Choquet Bonferroni mean operators by combining Choquet integral operators and Bonferroni mean operators based on Frank operations of BNNs. In the same study, Wang et al. [89] established an MCDM strategy based on Frank Choquet Bonferroni operators of BNNs in a bipolar neutrosophic environment. Pramanik et al. [90] developed a Tomada de decisao interativa e multicritévio (TODIM) strategy for MAGDM in a bipolar neutrosophic environment. An MADM strategy based on cross entropy for BNSs is yet to appear in the literature.
Mahmood et al. [91] and Deli et al. [92] introduced the hybridized structure called interval bipolar neutrosophic sets (IBNSs) by combining BNSs and INSs and defined some operations and operators for IBNSs. An MADM strategy based on cross entropy for IBNSs is yet to appear in the literature.
Research gap:
An MADM strategy based on cross entropy for BNSs and an MADM strategy based on cross entropy for IBNSs.
This paper answers the following research questions:
  • Is it possible to define a new cross entropy measure for BNSs?
  • Is it possible to define a new weighted cross entropy measure for BNSs?
  • Is it possible to develop a new MADM strategy based on the proposed cross entropy measure in a BNS environment?
  • Is it possible to develop a new MADM strategy based on the proposed weighted cross entropy measure in a BNS environment?
  • Is it possible to define a new cross entropy measure for IBNSs?
  • Is it possible to define a new weighted cross entropy measure for IBNSs?
  • Is it possible to develop a new MADM strategy based on the proposed cross entropy measure in an IBNS environment?
  • Is it possible to develop a new MADM strategy based on the proposed weighted cross entropy measure in an IBNS environment?
Motivation:
The above-mentioned analysis presents the motivation behind proposing a cross-entropy-based strategy for tackling MADM in BNS and IBNS environments. This study develops two novel cross-entropy-based MADM strategies.
The objectives of the paper are:
  • To define a new cross entropy measure and prove its basic properties.
  • To define a new weighted cross measure and prove its basic properties.
  • To develop a new MADM strategy based on the weighted cross entropy measure in a BNS environment.
  • To develop a new MADM strategy based on the weighted cross entropy measure in an IBNS environment.
To fill the research gap, we propose a cross-entropy-based MADM strategy in the BNS environment and the IBNS environment.
The main contributions of this paper are summarized below:
  • We propose a new cross entropy measure in the BNS environment and prove its basic properties.
  • We propose a new weighted cross entropy measure in the IBNS environment and prove its basic properties.
  • We develop a new MADM strategy based on weighted cross entropy to solve MADM problems in a BNS environment.
  • We develop a new MADM strategy based on weighted cross entropy to solve MADM problems in an IBNS environment.
  • Two illustrative numerical examples are solved and a comparison analysis is provided.
The rest of the paper is organized as follows. In Section 2, we present some concepts regarding SVNSs, INSs, BNSs, and IBNSs. Section 3 proposes cross entropy and weighted cross entropy measures for BNSs and investigates their properties. In Section 4, we extend the cross entropy measures for BNSs to cross entropy measures for IBNSs and discuss their basic properties. Two novel MADM strategies based on the proposed cross entropy measures in bipolar and interval bipolar neutrosophic settings are presented in Section 5. In Section 6, two numerical examples are solved and a comparison with other existing methods is provided. In Section 7, conclusions and the scope of future work are provided.

2. Preliminary

In this section, we provide some basic definitions regarding SVNSs, INSs, BNSs, and IBNSs.

2.1. Single-Valued Neutrosophic Sets

An SVNS [29] S in U is characterized by a truth membership function T S ( x ) , an indeterminate membership function I S ( x ) , and a falsity membership function F S ( x ) . An SVNS S over U is defined by
S = { x , T S ( x ) , I S ( x ) , F S ( x ) | x U }
where, T S ( x ) , I S ( x ) , F S ( x ) : U → [0, 1] and 0 T S ( x ) + I S ( x ) + F S ( x ) 3 for each point xU.

2.2. Interval Neutrosophic Set

An interval neutrosophic set [75] P in U is expressed as given below:
P = { x , T P ( x ) , I P ( x ) , F P ( x ) | x U } = { x , [ inf T p ( x ) ,   sup T p ( x ) ] ;   [ inf I p ( x ) ,   sup I p ( x ) ] ;   [ inf F p ( x )   sup F p ( x ) ] | x U }
where T P ( x ) , I P ( x ) , F P ( x ) are the truth membership function, indeterminacy membership function, and falsity membership function, respectively. For each point x in U, T P ( x ) , I P ( x ) , F P ( x ) [0, 1] satisfying the condition 0 sup T P ( x ) + sup I P ( x ) + sup F P ( x ) 3.

2.3. Bipolar Neutrosophic Set

A BNS [81] E in U is presented as given below:
E =   { x , T E + ( x ) , I E + ( x ) , F E + ( x ) , T E ( x ) ,   I E ( x ) ,   F E ( x ) | x U }
where T E + ( x ) , I E + ( x ) , F E + ( x ) : U → [0, 1] and T E ( x ) , I E ( x ) , F E ( x ) : U → [−1, 0]. Here, T E + ( x ) , I E + ( x ) , F E + ( x ) denote the truth membership, indeterminate membership, and falsity membership functions corresponding to BNS E on an element xU, and T E ( x ) , I E ( x ) , F E ( x ) denote the truth membership, indeterminate membership, and falsity membership of an element xU to some implicit counter property corresponding to E.
Definition 1.
Ref. [81]: Let, E1 = {x, T E 1 + ( x ) , I E 1 + ( x ) , F E 1 + ( x ) , T E 1 ( x ) ,   I E 1 ( x ) ,   F E 1 ( x ) |x ∈ U} and E2 = {x, T E 2 + ( x ) , I E 2 + ( x ) , F E 2 + ( x ) , T E 2 ( x ) ,   I E 2 ( x ) ,   F E 2 ( x ) |x ∈ X} be any two BNSs. Then
  • E1 E2 if, and only if,
T E 1 + ( x ) T E 2 + ( x ) , I E 1 + ( x ) I E 2 + ( x ) , F E 1 + ( x ) F E 2 + ( x ) ; T E 1 ( x ) T E 2 ( x ) , I E 1 ( x ) I E 2 ( x ) , F E 1 ( x ) F E 2 ( x ) for all x ∈ U.
  • E1 = E2 if, and only if,
T E 1 + ( x ) = T E 2 + ( x ) , I E 1 + ( x ) = I E 2 + ( x ) , F E 1 + ( x ) = F E 2 + ( x ) ; T E 1 ( x ) = T E 2 ( x ) , I E 1 ( x ) = I E 2 ( x ) , F E 1 ( x ) = F E 2 ( x ) for all x ∈ U.
  • The complement of E is Ec = {x, T E c + ( x ) , I E c + ( x ) , F E c + ( x ) , T E c ( x ) ,   I E c ( x ) ,   F E c ( x ) |x ∈ U}
where
T E c + ( x ) = F E + ( x ) , I E c + ( x ) = 1 I E + ( x ) , F E c + ( x ) = T E + ( x ) ;
T E c ( x ) = F E ( x ) , I E c ( x ) = 1 I E ( x ) , F E c ( x ) = T E ( x ) .
  • The union E1 E2 is defined as follows:
E1 E2 = {Max ( T E 1 + ( x ) , T E 2 + ( x ) ), Min ( I E 1 + ( x ) , I E 2 + ( x ) ), Min ( F E 1 + ( x ) , F E 2 + ( x ) ), Min ( T E 1 ( x ) , T E 2 ( x ) ), Max ( I E 1 ( x ) , I E 2 ( x ) ), Max ( F E 1 ( x ) , F E 2 ( x ) )}, x ∈ U.
  • The intersection E1 E2 [88] is defined as follows:
E1 E2 = {Min ( T E 1 + ( x ) , T E 2 + ( x ) ), Max ( I E 1 + ( x ) , I E 2 + ( x ) ), Max ( F E 1 + ( x ) , F E 2 + ( x ) ), Max ( T E 1 ( x ) , T E 2 ( x ) ), Min ( I E 1 ( x ) , I E 2 ( x ) ), Min ( F E 1 ( x ) , F E 2 ( x ) )}, x ∈ U.

2.4. Interval Bipolar Neutrosophic Sets

An IBNS [91,92] R = {x, <[inf T R + (x), sup T R + (x)]; [inf I R + (x), sup I R + (x)]; [inf F R + (x), sup F R + (x)]; [inf T R (x), sup T R (x)]; [inf I R (x), sup I R (x)]; [inf F R (x), sup F R (x)]>|xU} is characterized by positive and negative truth membership functions T R + (x), T R (x), respectively; positive and negative indeterminacy membership functions I R + (x), I R (x), respectively; and positive and negative falsity membership functions F R + (x), F R (x), respectively. Here, for any xU, T R + (x), I R + (x), F R + (x) [0, 1] and T R (x), I R (x), F R (x) [−1, 0] with the conditions 0 sup T R + (x) + sup I R + (x) + sup F R + (x) 3 and −3 sup T R (x) + sup I R (x) + sup F R (x) 0.
Definition 2.
Ref. [91,92]: Let R = {x, <[inf T R + (x), sup T R + (x)]; [inf I R + (x), sup I R + (x)]; [inf F R + (x), sup F R + (x)]; [inf T R (x), sup T R (x)]; [inf I R (x), sup I R (x)]; [inf F R (x), sup F R (x)]>|x ∈ U} and S = {x, <[inf T S + (x), sup T S + (x)]; [inf I S + (x), sup I S + (x)]; [inf F S + (x), sup F S + (x)]; [inf T S (x), sup T S (x)]; [inf I S (x), sup I S (x)]; [inf F S (x), sup F S (x)]>|x ∈ U} be two IBNSs in U. Then
  • R S if, and only if,
    inf T R + (x) inf T S + (x), sup T R + (x) sup T S + (x),
    inf I R + (x) inf I S + (x), sup I R + (x) sup I S + (x),
    inf F R + (x) inf F S + (x), sup F R + (x) sup F S + (x),
    inf T R (x) inf T S (x), sup T R (x) sup T S (x),
    inf I R (x) inf I S (x), sup I R (x) sup I S (x),
    inf F R (x) inf F S (x), sup F R (x) sup F S (x),
    for all x ∈ U.
  • R = S if, and only if,
    inf T R + (x) = inf T S + (x), sup T R + (x) = sup T S + (x), inf I R + (x) = inf I S + (x), sup I R + (x) = sup I S + (x),
    inf F R + (x) = inf F S + (x), sup F R + (x) = sup F S + (x), inf T R (x) = inf T S (x), sup T R (x) = sup T S (x),
    inf I R (x) = inf I S (x), sup I R (x) = sup I S (x), inf F R (x) = inf F S (x), sup F R (x) = sup F S (x),
    for all x ∈ U.
  • The complement of R is defined as The complement of R is defined as RC = {x, < [inf T R C + (x), sup T R C + (x)]; [inf I R C + (x), sup I R C + (x)]; [inf F R C + (x), sup F R C + (x)]; [inf T R C (x), sup T R C (x)]; [inf I R C (x), sup I R C (x)]; [inf F R C (x), sup F R C (x)] > | xU} where
    inf T R C + (x) = inf F R + (x), sup T R C + (x) = sup F R + (x)
    inf I R C + (x) = 1 − sup I R + (x), sup I R C + (x) = 1 − inf I R + (x)
    inf F R C + (x) = inf T R + , sup F R C + (x) = sup T R +
    inf T R C (x) = inf F R , sup T R C (x) = sup F R
    inf I R C (x) = −1 − sup I R (x), sup I R C (x) = −1 − inf I R (x)
    inf F R C (x) = inf T R (x), sup F R C (x) = sup T R (x)
    for all xU.

3. Cross Entropy Measures of Bipolar Neutrosophic Sets

In this section we define a cross entropy measure between two BNSs and establish some of its basic properties.
Definition 3.
For any two BNSs M and N in U, the cross entropy measure can be defined as follows.
C B ( M , N )   = i = 1 n [ T M + ( x i ) + T N + ( x i ) 2 ( T M + ( x i ) + T N + ( x i ) 2 ) + I M + ( x i ) + I N + ( x i ) 2 ( I M + ( x i ) + I N + ( x i ) 2 ) + ( 1 I M + ( x i ) ) + ( 1 I N + ( x i ) ) 2 ( ( 1 I M + ( x i ) ) + ( 1 I N + ( x i ) ) 2 ) + F M + ( x i ) + F N + ( x i ) 2 ( F M + ( x i ) + F N + ( x i ) 2 ) + ( T M ( x i ) + T N ( x i ) ) 2 ( ( T M ( x i ) ) + ( T N ( x i ) ) 2 ) + ( I M ( x i ) + I N ( x i ) ) 2 ( ( I M ( x i ) ) + ( I N ( x i ) ) 2 ) + ( 1 + I M ( x i ) ) + ( 1 + I N ( x i ) ) 2 ( 1 + I M ( x i ) + 1 + I N ( x i ) 2 ) + ( F M ( x i ) + F N ( x i ) ) 2 ( ( F M ( x i ) ) + ( F N ( x i ) ) 2 ) ]
Theorem 1.
If M = < T M + ( x i ) , I M + ( x i ) , F M + ( x i ) , T M ( x i ) , I M ( x i ) , F M ( x i ) > and N < T N + ( x i ) , I N + ( x i ) , F N + ( x i ) , T N ( x i ) , I N ( x i ) , F N ( x i ) > are two BNSs in U, then the cross entropy measure CB(M, N) satisfies the following properties:
(1)
CB(M, N) 0;
(2)
CB(M, N) = 0 if, and only if, T M + ( x i ) = T N + ( x i ) , I M + ( x i ) = I N + ( x i ) , F M + ( x i ) = F N + ( x i ) , T M ( x i ) = T N ( x i ) , I M ( x i ) = I N ( x i ) , F M ( x i ) = F N ( x i ) , x ∈ U;
(3)
CB(M, N) = CB(N, M);
(4)
CB(M, N) = CB(MC, NC).
Proof 
(1)
We have the inequality ( a + b 2 ) 1 2 a 1 2 + b 1 2 2 for all positive numbers a and b. From the inequality we can easily obtain CB(M, N) 0.
(2)
The inequality ( a + b 2 ) 1 2 a 1 2 + b 1 2 2 becomes the equality ( a + b 2 ) 1 2 = a 1 2 + b 1 2 2 if, and only if, a = b and therefore CB(M, N) = 0 if, and only if, M = N, i.e., T M + ( x i ) = T N + ( x i ) , I M + ( x i ) = I N + ( x i ) , F M + ( x i ) = F N + ( x i ) , T M ( x i ) = T N ( x i ) , I M ( x i ) = I N ( x i ) , F M ( x i ) = F N ( x i ) xU.
(3)
CB(M, N) = i = 1 n [ T M + ( x i ) + T N + ( x i ) 2 ( T M + ( x i ) + T N + ( x i ) 2 ) + I M + ( x i ) + I N + ( x i ) 2 ( I M + ( x i ) + I N + ( x i ) 2 ) + ( 1 I M + ( x i ) ) + ( 1 I N + ( x i ) ) 2 ( ( 1 I M + ( x i ) ) + ( 1 I N + ( x i ) ) 2 ) + F M + ( x i ) + F N + ( x i ) 2 ( F M + ( x i ) + F N + ( x i ) 2 ) + ( T M ( x i ) + T N ( x i ) ) 2 ( ( T M ( x i ) ) + ( T N ( x i ) ) 2 ) + ( I M ( x i ) + I N ( x i ) ) 2 ( ( I M ( x i ) ) + ( I N ( x i ) ) 2 ) + ( 1 + I M ( x i ) ) + ( 1 + I N ( x i ) ) 2 ( 1 + I M ( x i ) + 1 + I N ( x i ) 2 ) + ( F M ( x i ) + F N ( x i ) ) 2 ( ( F M ( x i ) ) + ( F N ( x i ) ) 2 ) ] = i = 1 n [ T N + ( x i ) + T M + ( x i ) 2 ( T N + ( x i ) + T M + ( x i ) 2 ) + I N + ( x i ) + I M + ( x i ) 2 ( I N + ( x i ) + I M + ( x i ) 2 ) + ( 1 I N + ( x i ) ) + ( 1 I M + ( x i ) ) 2 ( ( 1 I N + ( x i ) ) + ( 1 I M + ( x i ) ) 2 ) + F N + ( x i ) + F M + ( x i ) 2 ( F N + ( x i ) + F M + ( x i ) 2 ) + ( T N ( x i ) + T M ( x i ) ) 2 ( ( T N ( x i ) ) + ( T M ( x i ) ) 2 ) + ( I N ( x i ) + I M ( x i ) ) 2 ( ( I N ( x i ) ) + ( I M ( x i ) ) 2 ) + ( 1 + I N ( x i ) ) + ( 1 + I M ( x i ) ) 2 ( 1 + I N ( x i ) + 1 + I M ( x i ) 2 ) + ( F N ( x i ) + F M ( x i ) ) 2 ( ( F N ( x i ) ) + ( F M ( x i ) ) 2 ) ] = CB(N, M).
(4)
CB(MC, NC) = i = 1 n [ F M + ( x i ) + F N + ( x i ) 2 ( F M + ( x i ) + F N + ( x i ) 2 ) + ( 1 I M + ( x i ) ) + ( 1 I N + ( x i ) ) 2 ( ( 1 I M + ( x i ) ) + 1 I N + ( x i ) ) 2 ) + 1 ( 1 I M + ( x i ) ) + 1 ( 1 I N + ( x i ) ) 2 ( 1 ( 1 I M + ( x i ) ) + 1 ( 1 I N + ( x i ) ) 2 ) + T M + ( x i ) + T N + ( x i ) 2 ( T M + ( x i ) + T N + ( x i ) 2 ) + ( F M ( x i ) + F N ( x i ) ) 2 ( ( F M ( x i ) ) + ( F N ( x i ) ) 2 ) + ( 1 I M ( x i ) ) ( 1 I N ( x i ) ) 2 ( ( 1 I M ( x i ) ) + ( 1 I N ( x i ) ) 2 ) + 1 + ( 1 I M ( x i ) ) + 1 + ( 1 I N ( x i ) ) 2 ( 1 + ( 1 I M ( x i ) ) + 1 + ( 1 I N ( x i ) ) 2 ) + ( T M ( x i ) + T N ( x i ) ) 2 ( ( T M ( x i ) ) + ( T N ( x i ) ) 2 ) ] = i = 1 n [ T M + ( x i ) + T N + ( x i ) 2 ( T M + ( x i ) + T N + ( x i ) 2 ) + I M + ( x i ) + I N + ( x i ) 2 ( I M + ( x i ) + I N + ( x i ) 2 ) + ( 1 I M + ( x i ) ) + ( 1 I N + ( x i ) ) 2 ( ( 1 I M + ( x i ) ) + ( 1 I N + ( x i ) ) 2 ) + F M + ( x i ) + F N + ( x i ) 2 ( F M + ( x i ) + F N + ( x i ) 2 ) + ( T M ( x i ) + T N ( x i ) ) 2 ( ( T M ( x i ) ) + ( T N ( x i ) ) 2 ) + ( I M ( x i ) + I N ( x i ) ) 2 ( ( I M ( x i ) ) + ( I N ( x i ) ) 2 ) + ( 1 + I M ( x i ) ) + ( 1 + I N ( x i ) ) 2 ( 1 + I M ( x i ) + 1 + I N ( x i ) 2 ) + ( F M ( x i ) + F N ( x i ) ) 2 ( ( F M ( x i ) ) + ( F N ( x i ) ) 2 ) ] = CB(M, N).
The proof is completed. ☐
Example 1.
Suppose that M = <0.7, 0.3, 0.4, −0.3, −0.5, −0.1> and N = <0.5, 0.2, 0.5, −0.3, −0.3, −0.2> are two BNSs; then the cross entropy between M and N is calculated as follows:
C B ( M , N )   = [ 0.7 + 0.5 2 ( 0.7 + 0.5 2 ) + 0.3 + 0.2 2 ( 0.3 + 0.2 2 ) + ( 1 0.3 ) + ( 1 0.2 ) 2 ( 1 0.3 + 1 0.2 2 ) + 0.4 + 0.5 2 ( 0.4 + 0.5 2 ) + ( 0.3 0.3 ) 2 ( ( 0.3 ) + ( 0.3 ) 2 ) + ( 0.5 0.3 ) 2 ( ( 0.5 ) + ( 0.3 ) 2 ) + ( 1 0.5 ) + ( 1 0.3 ) 2 ( 1 0.5 + [ 1 0.3 ] 2 ) + ( 0.1 0.2 ) 2 ( ( 0.1 ) + ( 0.2 ) 2 ) ] = 0.01738474 .
Definition 4.
Suppose that wi is the weight of each element xi, i = 1, 2, ..., n, where wi [0, 1] and i = 1 n w i = 1; then the weighted cross entropy measure between any two BNSs M and N in U can be defined as follows.
C B ( M , N ) w = i = 1 n w i [ T M + ( x i ) + T N + ( x i ) 2 ( T M + ( x i ) + T N + ( x i ) 2 ) + I M + ( x i ) + I N + ( x i ) 2 ( I M + ( x i ) + I N + ( x i ) 2 ) + ( 1 I M + ( x i ) ) + ( 1 I N + ( x i ) ) 2 ( 1 I M + ( x i ) + 1 I N + ( x i ) 2 ) + F M + ( x i ) + F N + ( x i ) 2 ( F M + ( x i ) + F N + ( x i ) 2 ) + ( T M ( x i ) + T N ( x i ) ) 2 ( ( T M ( x i ) ) + ( T N ( x i ) ) 2 ) + ( I M ( x i ) + I N ( x i ) ) 2 ( ( I M ( x i ) ) + ( I N ( x i ) ) 2 ) + ( 1 + I M ( x i ) ) + ( 1 + I N ( x i ) ) 2 ( 1 + I M ( x i ) + 1 + I N ( x i ) 2 ) + ( F M ( x i ) + F N ( x i ) ) 2 ( ( F M ( x i ) ) + ( F N ( x i ) ) 2 ) ]
Theorem 2.
If M = < T M + ( x i ) , I M + ( x i ) , F M + ( x i ) , T M ( x i ) , I M ( x i ) , F M ( x i ) > and N < T N + ( x i ) , I N + ( x i ) , F N + ( x i ) , T N ( x i ) , I N ( x i ) , F N ( x i ) > are two BNSs in U, then the weighted cross entropy measure CB(M, N)w satisfies the following properties:
(1)
CB(M, N)w 0;
(2)
CB(M, N)w = 0 if, and only if, T M + ( x i ) = T N + ( x i ) , I M + ( x i ) = I N + ( x i ) , F M + ( x i ) = F N + ( x i ) , T M ( x i ) = T N ( x i ) , I M ( x i ) = I N ( x i ) , F M ( x i ) = F N ( x i ) , x ∈ U;
(3)
CB(M, N)w = CB(N, M)w;
(4)
CB(MC, NC)w = CB(M, N)w.
Proof is given in Appendix A.
Example 2.
Suppose that M = <0.7, 0.3, 0.4, −0.3, −0.5, −0.1> and N = <0.5, 0.2, 0.5, −0.3, −0.3, −0.2> are two BNSs and w = 0.4; then the weighted cross entropy between M and N is calculated as given below.
C B ( M , N ) w =   0.4 × [ 0.7 + 0.5 2 ( 0.7 + 0.5 2 ) + 0.3 + 0.2 2 ( 0.3 + 0.2 2 ) + ( 1 0.3 ) + ( 1 0.2 ) 2 ( 1 0.3 + 1 0.2 2 ) + 0.4 + 0.5 2 ( 0.4 + 0.5 2 ) + ( 0.3 0.3 ) 2 ( ( 0.3 ) + ( 0.3 ) 2 ) + ( 0.5 0.3 ) 2 ( ( 0.5 ) + ( 0.3 ) 2 ) + ( 1 0.5 + ( 1 0.3 ) 2 ( 1 0.5 + 1 0.3 2 ) + ( 0.1 0.2 ) 2 ( ( 0.1 ) + ( 0.2 ) 2 ) ] = 0.006953896 .

4. Cross Entropy Measure of IBNSs

This section extends the concepts of cross entropy and weighted cross entropy measures of BNSs to IBNSs.
Definition 5.
The cross entropy measure between any two IBNSs R = <[inf T R + ( x i ) , sup T R + ( x i ) ], [inf I R + ( x i ) , sup I R + ( x i ) ], [inf F R + ( x i ) , sup F R + ( x i ) ], [inf T R ( x i ) , sup T R ( x i ) ], [inf I R ( x i ) , sup I R ( x i ) ], [inf F R ( x i ) , sup F R ( x i ) ]> and S = <[inf T S + ( x i ) , sup T S + ( x i ) ], [inf I S + ( x i ) , sup I S + ( x i ) ], [inf F S + ( x i ) , sup F S + ( x i ) ], [inf T S ( x i ) , sup T S ( x i ) ], [inf I S ( x i ) , sup I S ( x i ) ], [inf F S ( x i ) , sup F S ( x i ) ]> in U can be defined as follows.
C I B ( R , S ) = 1 2 i = 1 n [ inf T R + ( x i ) + inf T S + ( x i ) 2 ( inf T R + ( x i ) + inf T S + ( x i ) 2 ) + sup T R + ( x i ) + sup T S + ( x i ) 2 ( sup T R + ( x i ) + sup T S + ( x i ) 2 ) + inf I R + ( x i ) + inf I S + ( x i ) 2 ( inf I R + ( x i ) + inf I S + ( x i ) 2 ) + sup I R + ( x i ) + sup I S + ( x i ) 2 ( sup I R + ( x i ) + sup I S + ( x i ) 2 ) + ( 1 inf I R + ( x i ) ) + ( 1 inf I S + ( x i ) ) 2 ( 1 inf I R + ( x i ) + [ 1 inf I S + ( x i ) ] 2 ) + ( 1 sup I R + ( x i ) ) + ( 1 sup I S + ( x i ) ) 2 ( 1 sup I R + ( x i ) + 1 sup I S + ( x i ) 2 ) + inf F R + ( x i ) + inf F S + ( x i ) 2 ( inf F R + ( x i ) + inf F S + ( x i ) 2 ) + sup F R + ( x i ) + sup F S + ( x i ) 2 ( sup F R + ( x i ) + sup F S + ( x i ) 2 ) + ( inf T R ( x i ) + inf T S ( x i ) ) 2 ( ( inf T R ( x i ) ) + ( inf T S ( x i ) ) 2 ) + ( sup T R ( x i ) + sup T S + ( x i ) ) 2 ( ( sup T R ( x i ) ) + ( sup T S ( x i ) ) 2 ) + ( inf I R ( x i ) + inf I S ( x i ) ) 2 ( ( inf I R ( x i ) ) + ( inf I S ( x i ) ) 2 ) + ( sup I R ( x i ) + sup I S ( x i ) ) 2 ( ( sup I R ( x i ) ) + ( sup I S ( x i ) ) 2 ) + ( 1 + inf I R ( x i ) ) + ( 1 + inf I S ( x i ) ) 2 ( 1 + inf I R ( x i ) + 1 + inf I S ( x i ) 2 ) + ( 1 + sup I R ( x i ) ) + ( 1 + sup I S ( x i ) ) 2 ( 1 + sup I R ( x i ) + 1 + sup I S ( x i ) 2 ) + ( inf F R ( x i ) + inf F S ( x i ) ) 2 ( ( inf F R ( x i ) ) + ( inf F S ( x i ) ) 2 ) + ( sup F R ( x i ) + sup F S ( x i ) ) 2 ( ( sup F R ( x i ) ) + ( sup F S ( x i ) ) 2 ) ]
Theorem 3.
If R = <[inf T R + ( x i ) , sup T R + ( x i ) ], [inf, sup I R + ( x i ) ], [inf F R + ( x i ) , sup F R + ( x i ) ], [inf T R ( x i ) , sup T R ( x i ) ], [inf I R ( x i ) , sup I R ( x i ) ], [inf F R ( x i ) , sup F R ( x i ) ]> and S = <[inf T S + ( x i ) , sup T S + ( x i ) ], [inf I S + ( x i ) , sup I S + ( x i ) ], [inf F S + ( x i ) , sup F S + ( x i ) ], [inf T S ( x i ) , sup T S ( x i ) ], [inf I S ( x i ) , sup I S ( x i ) ], [inf F S ( x i ) , sup F S ( x i ) ]> are two IBNSs in U, then the cross entropy measure CIB(R, S) satisfies the following properties:
(1)
CIB(R, S) 0;
(2)
CIB(R, S) = 0 for R = S i.e., inf T R + ( x i ) = inf T S + ( x i ) , sup T R + ( x i ) = sup T S + ( x i ) , inf I R + ( x i ) = inf I S + ( x i ) , sup I R + ( x i ) = sup I S + ( x i ) , inf F R + ( x i ) = inf F S + ( x i ) , sup F R + ( x i ) = sup F S + ( x i ) , inf T R ( x i ) = inf T S ( x i ) , sup T R ( x i ) = sup T S ( x i ) , inf I R ( x i ) = inf I S ( x i ) , sup I R ( x i ) = sup I S ( x i ) , inf F R ( x i ) = inf F S ( x i ) , sup F R ( x i ) = sup F S ( x i ) x ∈ U;
(3)
CIB(R, S) = CIB(S, R);
(4)
CIB(RC, SC) = CIB(R, S).
Proof 
(1)
From the inequality stated in Theorem 1, we can easily get CIB(R, S) 0.
(2)
Since inf T R + ( x i ) = inf T S + ( x i ) , sup T R + ( x i ) = sup T S + ( x i ) , inf I R + ( x i ) = inf I S + ( x i ) , sup I R + ( x i ) = sup I S + ( x i ) , inf F R + ( x i ) = inf F S + ( x i ) , sup F R + ( x i ) = sup F S + ( x i ) , inf T R ( x i ) = inf T S ( x i ) , sup T R ( x i ) = sup T S ( x i ) , inf I R ( x i ) = inf I S ( x i ) , sup I R ( x i ) = sup I S ( x i ) , inf F R ( x i ) = inf F S ( x i ) , sup F R ( x i ) = sup F S ( x i ) x ∈ U, we have CIB(R, S) = 0.
(3)
CIB(R, S) = 1 2 i = 1 n [ inf T R + ( x i ) + inf T S + ( x i ) 2 ( inf T R + ( x i ) + inf T S + ( x i ) 2 ) + sup T R + ( x i ) + sup T S + ( x i ) 2 ( sup T R + ( x i ) + sup T S + ( x i ) 2 ) + inf I R + ( x i ) + inf I S + ( x i ) 2 ( inf I R + ( x i ) + inf I S + ( x i ) 2 ) + sup I R + ( x i ) + sup I S + ( x i ) 2 ( sup I R + ( x i ) + sup I S + ( x i ) 2 ) + ( [ 1 inf I R + ( x i ) ) + ( 1 inf I S + ( x i ) ) 2 ( 1 inf I R + ( x i ) + 1 inf I S + ( x i ) 2 ) + ( 1 sup I R + ( x i ) ) + ( 1 sup I S + ( x i ) ) 2 ( 1 sup I R + ( x i ) + 1 sup I S + ( x i ) 2 ) + inf F R + ( x i ) + inf F S + ( x i ) 2 ( inf F R + ( x i ) + inf F S + ( x i ) 2 ) + sup F R + ( x i ) + sup F S + ( x i ) 2 ( sup F R + ( x i ) + sup F S + ( x i ) 2 ) + ( inf T R ( x i ) + inf T S ( x i ) ) 2 ( ( inf T R ( x i ) ) + ( inf T S ( x i ) ) 2 ) + ( sup T R ( x i ) + sup T S ( x i ) ) 2 ( ( sup T R ( x i ) ) + ( sup T S ( x i ) ) 2 ) + ( inf I R ( x i ) + inf I S ( x i ) ) 2 ( ( inf I R ( x i ) ) + ( inf I S ( x i ) ) 2 ) + ( sup I R ( x i ) + sup I S ( x i ) ) 2 ( ( sup I R ( x i ) ) + ( sup I S ( x i ) ) 2 ) + ( 1 + inf I R ( x i ) ) + ( 1 + inf I S ( x i ) ) 2 ( 1 + inf I R ( x i ) + 1 + inf I S ( x i ) 2 ) + ( 1 + sup I R ( x i ) ) + ( 1 + sup I S ( x i ) ) 2 ( 1 + sup I R ( x i ) + 1 + sup I S ( x i ) 2 ) + ( inf F R ( x i ) + inf F S ( x i ) ) 2 ( ( inf F R ( x i ) ) + ( inf F S ( x i ) ) 2 ) + ( sup F R ( x i ) + sup F S ( x i ) ) 2 ( ( sup F R ( x i ) ) + ( sup F S ( x i ) ) 2 ) ] = 1 2 i = 1 n [ inf T S + ( x i ) + inf T R + ( x i ) 2 ( inf T S + ( x i ) + inf T R + ( x i ) 2 ) + sup T S + ( x i ) + sup T R + ( x i ) 2 ( sup T S + ( x i ) + sup T R + ( x i ) 2 ) + inf I S + ( x i ) + inf I R + ( x i ) 2 ( inf I S + ( x i ) + inf I R + ( x i ) 2 ) + sup I S + ( x i ) + sup I R + ( x i ) 2 ( sup I S + ( x i ) + sup I R + ( x i ) 2 ) + ( 1 inf I S + ( x i ) ) + ( 1 inf I R + ( x i ) ) 2 ( 1 inf I S + ( x i ) + 1 inf I R + ( x i ) 2 ) + ( 1 sup I S + ( x i ) ) + ( 1 sup I R + ( x i ) ) 2 ( 1 sup I S + ( x i ) + 1 sup I R + ( x i ) 2 ) + inf F S + ( x i ) + inf F R + ( x i ) 2 ( inf F S + ( x i ) + inf F R + ( x i ) 2 ) + sup F S + ( x i ) + sup F R + ( x i ) 2 ( sup F S + ( x i ) + sup F R + ( x i ) 2 ) + ( inf T S ( x i ) + inf T R ( x i ) ) 2 ( ( inf T S ( x i ) ) + ( inf T R ( x i ) ) 2 ) + ( sup T S ( x i ) + sup T R ( x i ) ) 2 ( ( sup T S ( x i ) ) + ( sup T R ( x i ) ) 2 ) + ( inf I S ( x i ) + inf I R ( x i ) ) 2 ( ( inf I S ( x i ) ) + ( inf I R ( x i ) ) 2 ) + ( sup I S ( x i ) + sup I R ( x i ) ) 2 ( ( sup I S ( x i ) ) + ( sup I R ( x i ) ) 2 ) + ( 1 + inf I S ( x i ) ) + ( 1 + inf I R ( x i ) ) 2 ( 1 + inf I S ( x i ) + 1 + inf I R ( x i ) 2 ) + ( 1 + sup I S ( x i ) ) + ( 1 + sup I R ( x i ) ) 2 ( 1 + sup I S ( x i ) + [ 1 + sup I R ( x i ) ] 2 ) + ( inf F S ( x i ) + inf F R ( x i ) ) 2 ( ( inf F S ( x i ) ) + ( inf F R ( x i ) ) 2 ) + ( sup F S ( x i ) + sup F R ( x i ) ) 2 ( ( sup F S ( x i ) ) + ( sup F R ( x i ) ) 2 ) ] = CIB(S, R).
(4)
CIB(RC, SC) = 1 2 i = 1 n [ inf F R + ( x i ) + inf F S + ( x i ) 2 ( inf F R + ( x i ) + inf F S + ( x i ) 2 ) + sup F R + ( x i ) + sup F S + ( x i ) 2 ( sup F R + ( x i ) + sup F S + ( x i ) 2 ) + ( 1 inf I R + ( x i ) ) + ( 1 inf I S + ( x i ) ) 2 ( ( 1 inf I R + ( x i ) ) + ( 1 inf I S + ( x i ) ) 2 ) + ( 1 sup I R + ( x i ) ) + ( 1 sup I S + ( x ) i ) 2 ( ( 1 sup I R + ( x i ) ) + ( 1 sup I S + ( x i ) ) 2 ) + 1 ( 1 inf I R + ( x i ) ) + 1 ( 1 inf I S + ( x i ) ) 2 ( 1 ( 1 inf I R + ( x i ) ) + 1 ( 1 inf I S + ( x i ) ) 2 ) + 1 ( 1 sup I R + ( x i ) ) + 1 ( 1 sup I S + ( x i ) ) 2 ( 1 ( 1 sup I R + ( x i ) ) + 1 ( 1 sup I S + ( x i ) ) 2 ) + inf T R + ( x i ) + inf T S + ( x i ) 2 ( inf T R + ( x i ) + inf T S + ( x i ) 2 ) + sup T R + ( x i ) + sup T S + ( x i ) 2 ( sup T R + ( x i ) + sup T S + ( x i ) 2 ) + ( inf F R ( x i ) + inf F S ( x i ) ) 2 ( ( inf F R ( x i ) ) + ( inf F S ( x i ) ) 2 ) + ( sup F R ( x i ) + sup F S ( x i ) ) 2 ( ( sup F R ( x i ) ) + ( sup F S ( x i ) ) 2 ) + ( ( 1 inf I R ( x i ) ) + ( 1 inf I S ( x i ) ) ) 2 ( ( 1 inf I R ( x i ) ) + ( 1 inf I S ( x i ) ) 2 ) + ( ( 1 sup I R ( x i ) ) + ( 1 sup I S ( x i ) ) ) 2 ( ( 1 sup I R ( x i ) ) + ( 1 sup I S ( x i ) ) 2 ) + 1 + ( 1 inf I R ( x i ) ) + 1 + ( 1 inf I S ( x i ) ) 2 ( 1 + ( 1 inf I R ( x i ) ) + 1 + ( 1 inf I S ( x i ) ) 2 ) + 1 + ( 1 sup I R ( x i ) ) + 1 + ( 1 sup I S ( x i ) ) 2 ( 1 + ( 1 sup I R ( x i ) ) + 1 + ( 1 sup I S ( x i ) ) 2 ) + ( inf T R ( x i ) + inf T S ( x i ) ) 2 ( ( inf T R ( x i ) ) + ( inf T S ( x i ) ) 2 ) + ( sup T R ( x i ) + sup T S ( x i ) ) 2 ( ( sup T R ( x i ) ) + ( sup T S ( x i ) ) 2 ) ] = 1 2 i = 1 n [ inf T R + ( x i ) + inf T S + ( x i ) 2 ( inf T R + ( x i ) + inf T S + ( x i ) 2 ) + sup T R + ( x i ) + sup T S + ( x i ) 2 ( sup T R + ( x i ) + sup T S + ( x i ) 2 ) + inf I R + ( x i ) + inf I S + ( x i ) 2 ( inf I R + ( x i ) + inf I S + ( x i ) 2 ) + sup I R + ( x i ) + sup I S + ( x i ) 2 ( sup I R + ( x i ) + sup I S + ( x i ) 2 ) + ( [ 1 inf I R + ( x i ) ) + ( 1 inf I S + ( x i ) ) 2 ( 1 inf I R + ( x i ) + 1 inf I S + ( x i ) 2 ) + ( 1 sup I R + ( x i ) ) + ( 1 sup I S + ( x i ) ) 2 ( 1 sup I R + ( x i ) + 1 sup I S + ( x i ) 2 ) + inf F R + ( x i ) + inf F S + ( x i ) 2 ( inf F R + ( x i ) + inf F S + ( x i ) 2 ) + sup F R + ( x i ) + sup F S + ( x i ) 2 ( sup F R + ( x i ) + sup F S + ( x i ) 2 ) + ( inf T R ( x i ) + inf T S ( x i ) ) 2 ( ( inf T R ( x i ) ) + ( inf T S ( x i ) ) 2 ) + ( sup T R ( x i ) + sup T S ( x i ) ) 2 ( ( sup T R ( x i ) ) + ( sup T S ( x i ) ) 2 ) + ( inf I R ( x i ) + inf I S ( x i ) ) 2 ( ( inf I R ( x i ) ) + ( inf I S ( x i ) ) 2 ) + ( sup I R ( x i ) + sup I S ( x i ) ) 2 ( ( sup I R ( x i ) ) + ( sup I S ( x i ) ) 2 ) + ( 1 + inf I R ( x i ) ) + ( 1 + inf I S ( x i ) ) 2 ( 1 + inf I R ( x i ) + 1 + inf I S ( x i ) 2 ) + ( 1 + sup I R ( x i ) ) + ( 1 + sup I S ( x i ) ) 2 ( 1 + sup I R ( x i ) + 1 + sup I S ( x i ) 2 ) + ( inf F R ( x i ) + inf F S ( x i ) ) 2 ( ( inf F R ( x i ) ) + ( inf F S ( x i ) ) 2 ) + ( sup F R ( x i ) + sup F S ( x i ) ) 2 ( ( sup F R ( x i ) ) + ( sup F S ( x i ) ) 2 ) ] = CIB(R, S). ☐
Example 3.
Suppose that R = <[0.5, 0.8], [0.4, 0.6], [0.2, 0.6], [−0.3, −0.1], [−0.5, −0.1], [−0.5, −0.2]> and S = <[0.5, 0.9], [0.4, 0.5], [0.1, 0.4], [−0.5, −0.3], [−0.7, −0.3], [−0.6, −0.3]> are two IBNSs; the cross entropy between R and S is computed as follows:
C I B ( R , S )   = 1 2 [ 0.5 + 0.5 2 ( 0.5 + 0.5 2 ) + 0.8 + 0.9 2 ( 0.8 + 0.9 2 ) + 0.4 + 0.4 2 ( 0.4 + 0.4 2 ) + 0.6 + 0.5 2 ( 0.6 + 0.5 2 ) + [ 1 0.4 ] + [ 1 0.4 ] 2 ( 1 0.4 + [ 1 0.4 ] 2 ) + [ 1 0.6 ] + [ 1 0.5 ] 2 ( 1 0.6 + [ 1 0.5 ] 2 ) + 0.2 + 0.1 2 ( 0.2 + 0.1 2 ) + 0.6 + 0.4 2 ( 0.6 + 0.4 2 ) + ( 0.3 0.5 ) 2 ( ( 0.3 ) + ( 0.5 ) 2 ) + ( 0.1 0.3 ) 2 ( ( 0.1 ) + ( 0.3 ) 2 ) + ( 0.5 0.7 ) 2 ( ( 0.5 ) + ( 0.7 ) 2 ) + ( 0.1 0.3 ) 2 ( ( 0.1 ) + ( 0.3 ) 2 ) + [ 1 0.5 ] + [ 1 0.7 ] 2 ( 1 0.5 + [ 1 0.7 ] 2 ) + [ 1 0.1 ] + [ 1 0.3 ] 2 ( 1 0.1 + [ 1 0.3 ] 2 ) + ( 0.5 0.6 ) 2 ( ( 0.5 ) + ( 0.6 ) 2 ) + ( 0.2 0.3 ) 2 ( ( 0.2 ) + ( 0.3 ) 2 ) ] = 0.02984616 .
Definition 6.
Let wi be the weight of each element xi, i = 1, 2, ..., n, and wi [0, 1] with i = 1 n w i = 1; then the weighted cross entropy measure between any two IBNSs R = <[inf T R + ( x i ) , sup T R + ( x i ) ], [inf I R + ( x i ) , sup I R + ( x i ) ], [inf F R + ( x i ) , sup F R + ( x i ) ], [inf T R ( x i ) , sup T R ( x i ) ], [inf I R ( x i ) , sup I R ( x i ) ], [inf F R ( x i ) , sup F R ( x i ) ]> and S = <[inf T S + ( x i ) , sup T S + ( x i ) ], [inf I S + ( x i ) , sup I S + ( x i ) ], [inf F S + ( x i ) , sup F S + ( x i ) ], [inf T S ( x i ) , sup T S ( x i ) ], [inf I S ( x i ) , sup I S ( x i ) ], [inf F S ( x i ) , sup F S ( x i ) ]> in U can be defined as follows.
C I B ( R , S ) w = 1 2 i = 1 n w i [ inf T R + ( x i ) + inf T S + ( x i ) 2 ( inf T R + ( x i ) + inf T S + ( x i ) 2 ) + sup T R + ( x i ) + sup T S + ( x i ) 2 ( sup T R + ( x i ) + sup T S + ( x i ) 2 ) + inf I R + ( x i ) + inf I S + ( x i ) 2 ( inf I R + ( x i ) + inf I S + ( x i ) 2 ) + sup I R + ( x i ) + sup I S + ( x i ) 2 ( sup I R + ( x i ) + sup I S + ( x i ) 2 ) + ( 1 inf I R + ( x i ) ) + ( 1 inf I S + ( x i ) ) 2 ( 1 inf I R + ( x i ) + 1 inf I S + ( x i ) 2 ) + ( 1 sup I R + ( x i ) ) + ( 1 sup I S + ( x i ) ) 2 ( 1 sup I R + ( x i ) + ( 1 sup I S + ( x i ) ) 2 ) + inf F R + ( x i ) + inf F S + ( x i ) 2 ( inf F R + ( x i ) + inf F S + ( x i ) 2 ) + sup F R + ( x i ) + sup F S + ( x i ) 2 ( sup F R + ( x i ) + sup F S + ( x i ) 2 ) + ( inf T R ( x i ) + inf T S ( x i ) ) 2 ( ( inf T R ( x i ) ) + ( inf T S ( x i ) ) 2 ) + ( sup T R ( x i ) + sup T S ( x i ) ) 2 ( ( sup T R ( x i ) ) + ( sup T S ( x i ) ) 2 ) + ( inf I R ( x i ) + inf I S ( x i ) ) 2 ( ( inf I R ( x i ) ) + ( inf I S ( x i ) ) 2 ) + ( sup I R ( x i ) + sup I S ( x i ) ) 2 ( ( sup I R ( x i ) ) + ( sup I S ( x i ) ) 2 ) + ( 1 + inf I R ( x i ) ) + ( 1 + inf I S ( x i ) ) 2 ( 1 + inf I R ( x i ) + 1 + inf I S ( x i ) 2 ) + ( 1 + sup I R ( x i ) ) + ( 1 + sup I S ( x i ) ) 2 ( 1 + sup I R ( x i ) + [ 1 + sup I S ( x i ) ] 2 ) + ( inf F R ( x i ) + inf F S ( x i ) ) 2 ( ( inf F R ( x i ) ) + ( inf F S ( x i ) ) 2 ) + ( sup F R ( x i ) + sup F S ( x i ) ) 2 ( ( sup F R ( x i ) ) + ( sup F S ( x i ) ) 2 ) ]
Theorem 4.
For any two IBNSs R = <[inf T R + ( x i ) , sup T R + ( x i ) ], [inf I R + ( x i ) , sup I R + ( x i ) ], [inf F R + ( x i ) , sup F R + ( x i ) ], [inf T R ( x i ) , sup T R ( x i ) ], [inf I R ( x i ) , sup I R ( x i ) ], [inf F R ( x i ) , sup F R ( x i ) ]> and S = <[inf T S + ( x i ) , sup T S + ( x i ) ], [inf I S + ( x i ) , sup I S + ( x i ) ], [inf F S + ( x i ) , sup F S + ( x i ) ], [inf T S ( x i ) , sup T S ( x i ) ], [inf I S ( x i ) , sup I S ( x i ) ], [inf F S ( x i ) , sup F S ( x i ) ]> in U, the weighted cross entropy measure CIB(R, S)w also satisfies the following properties:
(1)
CIB(R, S)w 0;
(2)
CIB(R, S)w = 0 if, and only if, R = S i.e., inf T R + ( x i ) = inf T S + ( x i ) , sup T R + ( x i ) = sup T S + ( x i ) , inf I R + ( x i ) = inf I S + ( x i ) , sup I R + ( x i ) = sup I S + ( x i ) , inf F R + ( x i ) = inf F S + ( x i ) , sup F R + ( x i ) = sup F S + ( x i ) , inf T R ( x i ) = inf T S ( x i ) , sup T R ( x i ) = sup T S ( x i ) , inf I R ( x i ) = inf I S ( x i ) , sup I R ( x i ) = sup I S ( x i ) , inf F R ( x i ) = inf F S ( x i ) , sup F R ( x i ) = sup F S ( x i ) x ∈ U;
(3)
CIB(R, S)w = CIB(S, R)w;
(4)
CIB(RC, SC)w = CIB(R, S)w.
The proofs are presented in Appendix B.
Example 4.
Consider the two IBNSs R = <[0.5, 0.8], [0.4, 0.6], [0.2, 0.6], [−0.3, −0.1], [−0.5, −0.1], [−05, −0.2]> and S = <[0.5, 0.9], [0.4, 0.5], [0.1, 0.4], [−0.5, −0.3], [−0.7, −0.3], [−0.6, −0.3]>, and let w = 0.3; then the weighted cross entropy between R and S is calculated as follows:
C I B ( R , S )   = 1 2 × 0.3 × [ 0.5 + 0.5 2 ( 0.5 + 0.5 2 ) + 0.8 + 0.9 2 ( 0.8 + 0.9 2 ) + 0.4 + 0.4 2 ( 0.4 + 0.4 2 ) + 0.6 + 0.5 2 ( 0.6 + 0.5 2 ) + [ 1 0.4 ] + [ 1 0.4 ] 2 ( 1 0.4 + [ 1 0.4 ] 2 ) + [ 1 0.6 ] + [ 1 0.5 ] 2 ( 1 0.6 + [ 1 0.5 ] 2 ) + 0.2 + 0.1 2 ( 0.2 + 0.1 2 ) + 0.6 + 0.4 2 ( 0.6 + 0.4 2 ) + ( 0.3 0.5 ) 2 ( ( 0.3 ) + ( 0.5 ) 2 ) + ( 0.1 0.3 ) 2 ( ( 0.1 ) + ( 0.3 ) 2 ) + ( 0.5 0.7 ) 2 ( ( 0.5 ) + ( 0.7 ) 2 ) + ( 0.1 0.3 ) 2 ( ( 0.1 ) + ( 0.3 ) 2 ) + [ 1 0.5 ] + [ 1 0.7 ] 2 ( 1 0.5 + [ 1 0.7 ] 2 ) + [ 1 0.1 ] + [ 1 0.3 ] 2 ( 1 0.1 + [ 1 0.3 ] 2 ) + ( 0.5 0.6 ) 2 ( ( 0.5 ) + ( 0.6 ) 2 ) + ( 0.2 0.3 ) 2 ( ( 0.2 ) + ( 0.3 ) 2 ) ] = 0.00895385 .

5. MADM Strategies Based on Cross Entropy Measures

In this section, we propose two new MADM strategies based on weighted cross entropy measures in bipolar neutrosophic and interval bipolar neutrosophic environments. Let B = {B1, B2, …, Bm} (m 2) be a discrete set of m feasible alternatives which are to be evaluated based on n attributes C = {C1, C2, …, Cn} (n 2) and let wj be the weight vector of the attributes such that 0 wj 1 and j = 1 n w j = 1.

5.1. MADM Strategy Based on Weighted Cross Entropy Measures of BNS

The procedure for solving MADM problems in a bipolar neutrosophic environment is presented in the following steps:
Step 1. The rating of the performance value of alternative Bi (i = 1, 2, …, m) with respect to the predefined attribute Cj (j = 1, 2, …, n) can be expressed in terms of bipolar neutrosophic information as follows:
B i =   { C j ,   < T B i + ( C j ) , I B i + ( C j ) , F B i + ( C j ) , T B i ( C j ) , I B i ( C j ) , F B i ( C j ) > | C j C j , j =   1 ,   2 ,   , n } ,
where 0 T B i + ( C j ) + I B i + ( C j ) + F B i + ( C j ) 3 and −3 T B i ( C j ) + I B i ( C j ) + F B i ( C j ) 0, i = 1, 2, …, m; j = 1, 2, …, n.
Assume that d ˜ i j = < T i j + , I i j + , F i j + , T i j , I i j , F i j > is the bipolar neutrosophic decision matrix whose entries are the rating values of the alternatives with respect to the attributes provided by the expert or decision-maker. The bipolar neutrosophic decision matrix [ d ˜ i j ] m × n can be expressed as follows:
[ d ˜ i j ] m × n = B 1 B 2 . . B m C 1 C 2 C n ( d 11 d 12 d 1 n d 21 d 22 d 2 n . . . . . . d m 1 d m 2 d m n ) .
Step 2. The positive ideal solution (PIS) <p* = ( d 1 * , d 2 * , ..., d n * )> of the bipolar neutrosophic information is obtained as follows:
p j * = T j * + , I j * + , F j * + , T j * , I j * , F j * = < [ { M a x i ( T i j + ) | j   H 1 } ;   { M i n i ( T i j + ) | j H 2 } ] , [ { M i n i ( I i j + ) | j   H 1 } ;   { M a x i ( I i j + ) | j H 2 } ] , [ { M i n i ( F i j + ) | j H 1 } ;   { M a x i ( F i j + ) | j H 2 } ] , [ { M i n i ( T i j ) | j   H 1 } ;   { M a x i ( T i j ) | j H 2 } ] , [ { M a x i ( I i j ) | j   H 1 } ;   { M i n i ( I i j ) | j H 2 } ] , [ { M a x i ( F i j ) | j   H 1 } ;   { M i n i ( F i j ) | j H 2 } ] > ,   j   =   1 ,   2 ,   , n ;
where H1 and H2 represent benefit and cost type attributes, respectively.
Step 3. The weighted cross entropy between an alternative Bi, i = 1, 2, …, m, and the ideal alternative p* is determined by
C B ( B i , p * ) w = i = 1 n w i [ T i j + + T j * + 2 ( T i j + + T j * + 2 ) + I i j + + I j * + 2 ( I i j + + I j * + 2 ) + [ 1 I i j + ] + [ 1 I j * + ] 2 ( 1 I j + + [ 1 I j * + ] 2 ) + F i j + + F j * + 2 ( F i j + + F j * + 2 ) + ( T i j + T j * ) 2 ( ( T i j ) + ( T j * ) 2 ) + ( I i j + I j * ) 2 ( ( I i j ) + ( I j * ) 2 ) + [ 1 + I i j ] + [ 1 + I j * ] 2 ( 1 + I i j + [ 1 + I j * ] 2 ) + ( F i j + F j * ) 2 ( ( F i j ) + ( F j * ) 2 ) ] .
Step 4. A smaller value of CB(Bi, p*)w, i = 1, 2, ..., m represents that an alternative Bi, i = 1, 2, …, m is closer to the PIS p*. Therefore, the alternative with the smallest weighted cross entropy measure is the best alternative.

5.2. MADM Strategy Based on Weighted Cross Entropy Measures of IBNSs

The steps for solving MADM problems with interval bipolar neutrosophic information are presented as follows.
Step 1. In an interval bipolar neutrosophic environment, the rating of the performance value of alternative Bi (i = 1, 2, …, m) with respect to the predefined attribute Cj (j = 1, 2, …, n) can be represented as follows:
B i =   { C j ,   < [ inf T B i + ( C j ) ,   sup T B i + ( C j ) ] , [ inf I B i + ( C j ) , sup I B i + ( C j ) ] , [ inf F B i + ( C j ) ,   sup F B i + ( C j ) ] , [ inf T B i ( C j ) , sup T B i ( C j ) ] , [ inf I B i ( C j ) ,   sup I B i ( C j ) ] , [ inf F B i ( C j ) ,   sup F B i ( C j ) ] > | C j C j , j =   1 ,   2 ,   , n }
where 0 sup T B i + ( C j ) + sup I B i + ( C j ) + sup F B i + ( C j ) 3 and −3 sup T B i ( C j ) + sup I B i ( C j ) + sup F B i ( C j ) 0; j = 1, 2, …, n. Let g ˜ i j = <[ T L i j + , T U i j + ], [ I L i j + , I U i j + ], [ F L i j + , F U i j + ], [ T L i j , T U i j ], [ I L i j , I U i j ], [ F L i j , F U i j ]> be the bipolar neutrosophic decision matrix whose entries are the rating values of the alternatives with respect to the attributes provided by the expert or decision-maker. The interval bipolar neutrosophic decision matrix [ g ˜ i j ] m × n can be presented as follows:
[ g ˜ i j ] m × n = B 1 B 2 . . B m C 1 C 2 C n ( g 11 g 12 g 1 n g 21 g 22 g 2 n . . . . . . g m 1 g m 2 g m n ) .
Step 2. The PIS <q* = ( g 1 * , g 2 * , ..., g n * )> of the interval bipolar neutrosophic information is obtained as follows:
q j * = < [ T L i j * + , T U i j * + ] , [ I L i j * + , I U i j * + ] , [ F L i j * + , F U i j * + ] , [ T L i j * , T U i j * ] , [ I L i j * , I U i j * ] , [ F L i j * , F U i j * ] > , = < [ { M a x i ( T L i j + ) | j H 1 } ; { M i n i ( T L i j + ) | j H 2 } , { M a x i ( T U i j + ) | j H 1 } ; { M i n i ( T U i j + ) | j H 2 } ] , [ { M i n i ( I L i j + ) | j H 1 } ; { M a x i ( I L i j + ) | j H 2 } , { M i n i ( I U i j + ) | j H 1 } ; { M a x i ( I U i j + ) | j H 2 } ] , [ { M i n i ( F L i j + ) | j H 1 } ; { M a x i ( F L i j + ) | j H 2 } , { M i n i ( F U i j + ) | j H 1 } ; { M a x i ( F U i j + ) | j H 2 } ] , [ { M i n i ( T L i j ) | j H 1 } ; { M a x i ( T L i j ) | j H 2 } , { M i n i ( T U i j ) | j H 1 } ; { M a x i ( T U i j ) | j H 2 } ] , [ { M a x i ( I L i j ) | j H 1 } ; { M i n i ( I L i j ) | j H 2 } , { M a x i ( I U i j ) | j H 1 } ; { M i n i ( I U i j ) | j H 2 } ] , [ { M a x i ( F L i j ) | j H 1 } ; { M i n i ( F L i j ) | j H 2 } , { M a x i ( F U i j ) | j H 1 } ; { M i n i ( F U i j ) | j H 2 } ] > ,   j   =   1 ,   2 ,   , n
where H1 and H2 stand for benefit and cost type attributes, respectively.
Step 3. The weighted cross entropy between an alternative Bi, i = 1, 2, …, m, and the ideal alternative q* under an interval bipolar neutrosophic setting is computed as follows:
C I B ( B i , q * ) w = 1 2 × i = 1 n w i [ T L i j + + T L j * + 2 ( T L i j + + T L j * + 2 ) + T U i j + + T U j * + 2 ( T U i j + + T U j * + 2 ) + I L i j + + I L j * + 2 ( I L i j + + I L j